Representations of Semisimple Lie Algebras in the BGG Category O

This book is devoted mainly to a study of properties of the so-called category O of g-modules, where g is a complex finite dimensional semisimple Lie algebra. The category O was introduced in the 70s by I. N. Bernstein, I. M. Gel'fand and S. I. Gel'fand and study of its properties soon led to the famous Kazhdan-Lusztig conjecture and its beautiful proof by methods of algebraic analysis and algebraic geometry. In the first part of the book, attention is concentrated to the highest weight modules (category O, basic character formulae, BGG reciprocity, the BGG description of composition factors for Verma modules, the BGG resolution of finite-dimensional modules, translation functors, Schubert varieties and the Kazhdan-Lusztig polynomials).

The second part of the book is more advanced. It starts with a treatment of the relative (parabolic) version of the previous theory and then it presents a discussion of projective functors and their connection to translation functors and principal series modules, the role of tilting modules, fusion rules and formal characters, and shuffling, twisting and completion functors and their comparison. The last chapter presents a rapid survey of many recent topics (with many references), including classification of primitive ideals of the universal enveloping algebra, highest weight categories for algebras of other types (e.g. for Kac-Moody algebras, Virasoro algebra and quantised enveloping algebras), quivers and the Kozsul duality. The book is written in a precise and understandable style and it should be very helpful for anybody interested in representation theory.